Comparison between the new mechanistic and the chaos scale-up methods for gas-solid fluidized beds

Haidar Taofeeq ,Muthanna Al-Dahhan *

1 Multiphase Reactors Engineering and Applications Laboratory(mReal),Department of Chemical&Biochemical Engineering,Missouri University of Science&Technology,Rolla,MO 65409,USA

2 Chemical Engineering Department,College of Engineering,Al-Nahrain University,Baghdad,Iraq

3 Cihan University,Erbil,Iraq

4 Prosthetics and Orthotics Engineering Department,College of Engineering,Al-Nahrain University,Baghdad,Iraq

Keywords:Scale-up Chaotic analysis approach New scale-up methodology Kolmogorov entropy Gas–solid fluidized bed

A B S T R A C T The chaotic scale-up approach by matching the Kolmogorov entropy(E K)proposed by Schouten et al.(1996)was assessed in two geometrically similar gas–solid fluidized bed columns of 0.14 and 0.44 m diameter.We used four conditions of our validated new mechanistic scale-up method based on matching the radial profiles of gas holdup where the local dimensionless hydrodynamic parameters were similar as measured by advanced measurement techniques.These experimental conditions were used to evaluate the validity of the chaotic scale-up method,which were selected based on our new mechanistic scale-up methodology.Pressure gauge transducer measurements at the wall and inside the bed at various local radial locations and at three axial heights were used to estimate KE.It was found that the experimental conditions with similar or close radial profiles of the Kolmogorov entropy and with similar or close radial profiles of the gas holdup achieve the similarity in local dimensionless hydrodynamic parameters,and vice versa.

1.Introduction

The fluidized bed is considered one of the most important solid–gas reaction and contacting systems with a vast number of industrial applications,such as catalyst regeneration,drying,catalytic cracking,Fischer–Tropsch synthesis,and gas–solid polymerization[1,2].Gas–solid fluidized bed reactors are characterized by many advantages compared with the other types of reactors(e.g., fixed bed reactors)which include simple construction,relatively low operating and maintenance expenses,low pressure drop,approximately isothermal temperature distribution,excellent contact and good mixing between the gas and solid particles,good mass and heat transfer rates,and the ability to handle a large quantity of solid particles even with a continuous process rate[3].

Despite all these advantages,due to the complexity of flow structure and multifaceted interaction between the phases of gas–solid fluidized beds,it has been challenging to understand and quantify their hydrodynamics,design,scale-up,and performance.In addition,the gas–solid mixing behavior is poorly understood[4].These drawbacks make it difficult to scale up gas–solid fluidized bed reactors from small-scale(laboratory-or pilot plant-scale)to industrial-scale.Rüdisüli et al.[2]reported some of the pitfalls that could be associated with poor scale-up,such as gas bypassing,gas channeling,partial defluidization,erosion and damage to immersed surfaces,elutriation of solid particles,reduction in the heat and mass transfer rate performance,and insufficient solid particle mixing.

Many experimental and numerical studies related to scale-up of gas–solid fluidized beds have been reported in the open literature[5–8].Asa result,various scaling methods have been proposed to maintain hydrodynamic similarity in scaling up of the gas–solid fluidizing beds[6–8].These scale-up methods for geometrically similar gas–solid fluidized beds can be characterized as follows:(1)matching key dimensionless groups[3,9–13],(2)matching chaotic behavior by estimating Kolmogorov entropy(EK)of the pressure signal to describe the order/disorder of the system[14–16],and(3)matching the radial or diameter profiles of the gas holdups as a mechanistic new method since the gas phase dictates the dynamics of these beds[6–8,17].

In our research group,we have assessed the scaling up method based on matching dimensionless groups using advanced measurement techniques of optical fiber probe utilization,radioactive particle tracking(RPT),gamma ray computed tomography(CT),and gamma ray densitometry(GRD).We found that the used dimensionless groups are not sufficient to maintain hydrodynamic similarity and it will become difficult to apply if the number of the dimensionless groups to be matched increases[6–8].Al-Dahhan et al.[17]proposed a new mechanistic methodology for scaling up gas–solid fluidized beds to achieve hydrodynamic similarity among geometrically similar beds.This method is based on matching the radial profiles of the gas phase holdup at a height within the bed that could represent the hydrodynamics of the bed.Advanced measurement techniques have been used to validate this method by measuring local detailed hydrodynamics using optical fiber probe,gamma ray computed tomography(CT),radioactive particle tracking(RPT),and gamma ray densitometry(GRD)techniques[6–8].How ever,the method that is based on matching Kolmogorov entropy(EK)that was proposed by Schouten et al.[14]of the pressure signal measured at the wall has not been evaluated by measuring the detailed local hydrodynamic parameters using the above-mentioned techniques.Schouten et al.[14]proposed matching Kolmogorov entropy(EK)estimated from the pressure drop signal measured at the wall to scale-up and maintain hydrodynamic similarity of gas–solid fluidized beds.In this case,KE represents the degree of freedom of the system or in other words the degree of the order/disorder behavior of the system.The basic concept of this chaos analysis based method is that the rate of information loss should be kept similar when scaling up a fluidized bed from a small-scale to the large-scale,to ensure the hydrodynamic similarity between the two scaled beds.The advantage of this method as stated by Schouten et al.[14]is that the KE is explicitly linked to the bed diameter and hence the same solid particles can be used in both scales of the fluidized beds.Thus,the problem of finding appropriate solid particles is averted as in the case of matching dimensionless groups.In addition,the dimensionless entropy group number(EK.dp/u)is directly proportional to the Froude number(ug2/gdp)and the ratio between the static bed height and the bed diameter.van den Bleek and Schouten[15,16]claimed that when the dimensionless entropy group number is matched in the two scaled fluidized beds,the matching of dimensionless scaling groups in terms of the Froude number and H/Dcratio are enough to have the cases matching.

Accordingly,the focus of this work is to assess the scale-up of a gas–solid fluidized bed based on the chaos analysis based methodology proposed by Schouten et al.[14],by applying their methodology using pressure signal on the matching cases using our new mechanistic scale-up methodology,which is based on matching the radial profiles of the gas holdups between two fluidized beds.As well,the similarity detailed hydrodynamic parameters have been measured and confirmed using the above mentioned advanced measurement techniques.In this case,at these conditions we will assess if the estimated KE from the measured pressure signal at the wall and inside the bed at various axial and radial locations are matched or not.

2.Assessment of the Chaotic Method for Scale-up of Fluidized Bed

The chaotic based scale-up methodology was assessed using the experimental conditions that we used for validating our new mechanistic scale-up methodology,that is based on matching the radial profiles of the gas holdup between two scales of gas–solid fluidized beds that are geometrically similar.Therefore,the experimental conditions used by Zaid[6],Efhaima[7],and Efhaima and Al-Dahhan[8]were used in the present study,as illustrated in Table 1.In this table,there are conditions of Case B with respect to the conditions of the reference case(Case A)that provide similar gas holdup radial profiles as we confirmed and measured by optical fiber probe and gamma ray computed tomography(CT)measurements in these two beds.The local hydrodynamic parameters such as dimensionless solid velocity,gas/solid holdups,and dimensionless turbulent parameters(stresses and turbulent kinetic energy)have been measured using radioactive particle tracking,gamma ray computed tomography and optical fiber probe techniques.We found that these hydrodynamic parameters are similar or close to each other when the radial profiles of the gas holdup are close to each other.The question then will be whether the Kolmogorov entropy(EK)of the pressure signal measured at the wall or inside the bed be similar or close to each other or not in these bedsidentical to Cases A and B.This has been assessed here by adopting the conditions of Case A and the conditions of Case B for similar(εg,r).Since we have already approved the similarity of these mentioned local parameters that have been reported in Zaid[6],Al-Dahhan et al.[17],Efhaima[7],and Efhaima and Al-Dahhan[8],we are not going to report these results rather that we state that if KEs are similar or not when these local hydrodynamic parameters are similar and vice versa.The same approach will be applied to the cases where the hydrodynamic parameters are not similar which are for the cases of Case C and Case D with respect to the reference Case A.

Table 1Conditions that provide similar gas holdup radial profiles giving similarity in local hydrodynamics and non-similar gas holdup radial profiles giving non-similarity in local hydrodynamics

In this approach,Case A was selected as a reference condition,while Case B was identified(matching conditions)to have similar or close radial profiles of the gas holdup.Cases C and D were selected as mismatching conditions because they have different radial profiles of radial gas holdup compared with the reference condition(Case A).It is worth mentioning that the new scale-up methodology was validated using both invasive and noninvasive techniques mentioned above.We confirmed that Cases A and B have the same radial profiles of dimensionless particle velocity in the form of(Vp/umf),where umfis the minimum fluidization velocity.Additionally,the radial profiles of the dimensionless turbulent parameters with respect to the minimum fluidization velocities(e.g.,dimensionless shear stresses,dimensionless turbulent kinetic energy,and dimensionless eddy diffusivity)were matched for Cases A and B[6–8,17].

3.Experimental Setup

The experimental setup consisted of two fluidized bed columns of 0.14 m and 0.44 m inside diameters,with similar geometries.Both columns were constructed from Plexiglas?,and the plenums were made of aluminum.The columns and plenums were placed on the top of a stainless steel base.An industrial-scale compressor was used to supply compressed air to the columns at pressures up to 1.38 MPa.Omega flow meters(Omega Inc.,model FL-6715A)controlled the gas flow rateentering the columns.Schematic diagrams of the two fluidized bed columns are shown in Figs.1 and 2.

The 0.14 m inside diameter column was 1.84 m high and connected at its top with an upper section that had a larger diameter of 0.42 m and 0.84 m height to disengage the solid particles from the flowing gas by reducing the superficial gas velocity and hence the particle velocity.The gas phase was introduced through a sparger tube inside the plenum section and then through a distributor plate affixed between the column and plenum sections.The gas distributor plate was manufactured of a porous polyethylene sheet and had a pore size of 15–40 μm with about 1%opening area.The plate thickness is 0.0127 m which is strong enough to support the mass of the loaded solid particles and tolerate the inlet gas pressure.The sparger tube was plugged at one end and had 14 holes,all facing downward.This sparger construction makes the gas distribution more homogenous.

Fig.1.Schematic diagram of 0.14 m inside diameter fluidized bed column.

The 0.44 m inside diameter fluidized bed column closely resembled the 0.14 m inside diameter fluidized bed column.The shape of the upper section was similar,but it had an inside diameter of 0.88 m and was 0.95 m high.The distributor design was also similar to that used in the 0.14 m diameter fluidized bed column,and the plenum consisted of a sparger tube,which had 20 holes,all facing downwards.Both fluidized bed columns were electrically grounded to minimize the electrostatic effects.A photo of the two fluidized bed columns is shown in Fig.3.

The gauge-pressure transducer measurements were acquired at H/Dc=0.75,1.5,and 1.75 above the gas distributor for both fluidized bed columns.The selection of three axial heights was made to cover three important axial zones inside the fluidized bed:(1)H/Dc=0.75,which represents the axial zone near the distributor plate,when the bubbles first form and rise through the dense phase;(2)H/Dc=1.5,which represents the axial zone that is located approximately in the middle of the fluidizing bed which is the region that represents the bed hydrodynamics;and(3)H/Dc=1.75,which represents the axial zone near the freeboard of the column,when the bubbles and their wakes start to disengage and leave the bed.

In addition,local measurements using a tube connected to the pressure transducer and it is called here the pressure probe were taken at six radial positions(r/R=0.0,0.2,0.4,0.6,0.8,and 1.0)and at the same mentioned H/Dc,as shown in Fig.4.

The solid particles used in this work were similar to the cases listed in Table 1 and were glass beads with two average particle sizes(70 μm and 210 μm)and a particle density of 2500 kg·m?3.

Fig.2.Schematic diagram of 0.44 m ID fluidized bed column.

4.Experimental Technique

4.1.Single-ended pressure transducer

A single-ended pressure transducer(Omega Inc.,model PX-409-050GV)was used to measure the pressure fluctuation signals at three axial heights and six radial positions of the fluidized beds mentioned before,covering the gauge pressure range from 0 to 345 kPa.The signal was received by the data acquisition(DAQ)system(Omega Inc.,model OMB-DAQ-3000),which has high-speed capability in collecting data up to 106Hz,and was connected to the computer.The signals were recorded for 100 s at a rate of 100 Hz and repeated five times to ensure the reproducibility of the results,and the standard deviation was found to be less than 5%.The margins of error bars in terms of standard deviation were found smaller than the size of data point symbols.It is worthy to mention that a wide range of sampling frequency(25 to 500 Hz)was used to estimate which sampling rate properly provides the Kolmogorov entropy KE since it is highly dependent on the sampling rate as stated by van Ommen et al.[18,19].The number of data points for each signal was 104,as recommended by Schouten et al.[14]to be an adequate measurement of the KE estimation.

4.2.Local pressure probe connected to the pressure transducer

A local pressure probe of 2.5 mm inside diameter and lengths of 0.2 m and 0.3 m tubes made from stainless steel were connected to a single-ended pressure transducer to measure the local pressure fluctuations at a number of radial and axial locations inside the used fluidized bed as shown in Fig.5.The local pressure probe of the 0.2 m length tube was used for the column of 0.14 m diameter,while the probe of the 0.3 m length tube was used for the column of 0.44 m diameter.The inside diameter of the probes was chosen to ensure that the pressure fluctuation signals were collected without any damping due to the small inside diameter of the probe(which was reported to be less than 2 mm)or any resonance that could occur as a result of using a probe with a large inside diameter(which was reported to be higher than 5 mm),as recommended by van Ommen et al.[18,20,21].The end tips of the probes were covered with a wire mesh to prevent solid particles from entering the probes and blocking the tips or damping the pressure transducer,which would disturb the measurements.The wire mesh was stainless steel,with a 30 μm mesh diameter and 20 μm wire diameter and the opening fraction of 36%which had no considerable effect on the pressure fluctuations[20].Typical raw pressure fluctuation signals are illustrated in Fig.6 for different cases(A,B,C,and D)at r/R=0 and H/D=1.5.It is obvious that the signature of pressure fluctuation signals for Cases A and B is close to each other,while it is different between Cases A and C and between Cases A and D.This is also reflected in the magnitudes of the absolute relative difference(ARD)between the KE of these cases presented in Fig.7 that will be discussed later.

Fig.3.Photo of the two fluidized bed columns.

Fig.4.Local measurements at six radial positions for all three heights of both columns.

Fig.5.The local pressure probe connected to a single-ended pressure transducer used in a 0.14 m inside diameter fluidized bed column.

5.Outline KE Estimation

Fluidized beds have been characterized with dynamic behavior that is considered chaotic.The chaotic characteristics of these types of reactors results from the complex interaction between the gas phase and its surroundings(e.g.,solid particles,the vessel wall,and/or internals inside the bed).The degree of freedom of a chaotic system of gas–solid fluidized beds can be affected by many parameters,such as operating conditions,design parameters,and the physicochemical properties of gas and solid particles.Consequently,the rate of the loss of information inside fluidized beds which represents the degree of order and disorder of the dynamic system is a function of many hydrodynamic parameters,such as voidage,solid velocity and turbulent parameters,bubble size,bubble rise velocity,and bubble frequency.Many analytical methods have been used to represent the chaotic degree or the chaos state of a gas–solid fluidized bed,such as attractor reconstruction,correlation dimension,entropy,and the Kolmogorov entropy(EK).van Ommen et al.[18]showed that the Kolmogorov entropy is considered the most appropriate way to explain the chaotic degree of gas–solid fluidization systems,because it is easy to calculate and the analysis of the time series of the pressure fluctuation using the KE gives a clear picture about the chaos behavior of the system.Hence,KE becomes the obvious choice for estimating the chaotic degree or loss of information in gas–solid fluidized beds.

Fig.6.The typical raw pressure fluctuation signals for all cases at H/D c=1.5 and r/R=0.0.

Additionally,it has been found that the Kolmogorov entropy is considered a useful tool for identifying and distinguishing the flow regimes and their transition velocities in gas–solid fluidized systems and other multiphase flow s.It is also known that flow regimes play an important role in the scale-up process because they identify the way that both solids and gases interact inside the bed under different operating and design conditions[18,22–26].In this study,the method used to calculate KE is based on the maximum likelihood estimation of entropy proposed by Schouten et al.[27]and used by Nedeltchev et al.[25,26]:

The variable birepresents the number of sequential pairs of pointsin the time series signal of the pressure fluctuations.Each selected pair of points is compared with the rest of the points of the time series.In this case,the initial pair of the points is the starting pair[19].The following pair is selected after a length of lo(maximum interpoint distance),which is determined by lo=3×AAD.The ADD is the average absolute deviation of the time series signal.More details about the calculation of KE using the maximum likelihood estimation of entropy can be found in Schouten et al.[27].

6.Results and Discussion

The statistical differences in the measurements of KE profiles between the conditions illustrated in Table 1 are represented in terms of the percentage change in the average absolute relative difference(AARD)of all the local measurements and the percentage change in the absolute relative difference(ARD)of each individual local point as follow s:

w here x and y are the measured local Kolmogorov entropies at the radial locations(r)for the cases outlined in Table 1,and N is the total number of the local data points.The reproducibility of the experiments is one of themost crucial factors to consider before taking any measurements.To check the reproducibility of the pressure fluctuation,measurements were repeated five times at each local position for each experimental condition.The local averaged Kolmogorov entropy values were almost identical with few differences which were within about a 5%margin of difference.The error bars shown in each figure represent the standard deviation around the mean and they were found to be within the data points.

6.1.Radial profiles of KE for matching cases(Cases A and B)

Fig.7.Radial variations of the ARD in the Kolmogorov entropy for All Cases at different axial heights(a)H/D c=0.75,(b)H/D c=1.5,and(c)H/D c=1.75.The left side is for the Cases Aand B of similar hydrodynamics with matching radial profiles of gas holdup,the middle side is for the Cases A and C of non-similar hydrodynamics with mismatching radial profiles of gas holdup,and the right side is for the Cases A and D of non-similar hydrodynamics with mismatching radial profiles of gas holdup.

Fig.8 illustrates the radial profiles of the Kolmogorov entropy of the pressure signal measured at several radial locations and three axial heights for the experimental conditions of matching hydrodynamics of Cases A and B as listed in Table 1.As shown in the figure,the radial profiles of KE for Cases A and B were close or similar for all axial and radial locations within the bed.The percentage change in the average absolute relative difference(AARD)was4.7%at H/Dc=0.75.The results were not much different when H/Dcchanged from 0.75 to 1.5.The percentage change in the average absolute relative difference(AARD)was about 5.5%at H/Dc=1.5.The same trend of similar or close radial profiles of the Kolmogorov entropy was obtained at H/Dc=1.75,where the percentage change in the average absolute relative difference(AARD)was about 2.5%at H/Dc=0.75.The radial variations of the percentage change in the absolute relative difference(ARD)of KE between Cases A and B at three axial heights above the distributor are shown in Fig.7(left column),where the trends in the radial variations of the ARD were generally the same at different axial levels,and the local values of the ARD variation,as well,the value of ARD decreases with an increase in the ratio H/Dc.It is worthy to mention that the ARD values are relatively larger within the range of r/R=0.4–0.8 and this is could be due to the inversion point of the time averaged solid velocity from positive to negative values which occur at about r/R=0.65–0.68,where the fluctuations at these points need to be recorded for a longer time or due to the nature of such local locations.How ever,at the wall,the ARD is also low for all axial measured locations as shown in Fig.7.The differences attained between the studied cases are reasonable which indicates also the similarity in the chaotic behavior between Cases A and B where the local dimensionless hydrodynamic parameters are similar or close[6-8,11].Therefore,we can conclude that similar to our validated mechanistic scale-up methodology based on matching radial profiles of gas holdup,when KE of the pressure signal measured at the wall or inside the bed of two scales is maintained similar or close to each other,the local hydrodynamic similarity in terms of dimensionless parameters is similar or close to each other in the targeted two scales.This is an important finding since pressure fluctuation measurement at thewall is easier to implement as compared to the measurement of radial profiles of the gas holdup which are needed for our mechanistic scale-up methodology.Hence,the chaos analysis of pressure fluctuations at the wall using KE would be more easy for industrial scale-up applications.In this case,the industrial application of the chaos analysis for scale-up will be applied to any reactor size provided that the operating and design conditions of geometrically similar reactors should be selected to provide similar or close to the KE of the pressure fluctuation signal measured at the wall.

6.2.Radial profiles of KE for Cases A,C,and D

The radial profiles of KE for the cases of the experimental conditions(either Case C or D)that have non-similar radial profiles of gas holdup with respect to the reference case(Case A)are demonstrated in Figs.9 and 10,respectively.For these conditions,the local hydrodynamics are not similar as reported earlier and the details can be found in Zaid[6],Al-Dahhan et al.[17],Efhaima[7],and Efhaima and Al-Dahhan[8].Figs.9 and 10 show a large difference between the radial profiles of the Kolmogorov entropy for the non-similar cases(Cases C and D)with respect to the reference case as compared to Fig.8.These differences are comparable with the differences in gas holdup profiles and hence in the local hydrodynamic parameters.How ever,the radial profiles of the Kolmogorov entropy at H/Dc=1.75 are close,as shown in Figs.9 and 10.This may be caused by the measurements being close to the free board of the column where the gas phase starts to disengage from the bed.In the free board and gas disengagement regions,the chaotic behavior could be similar for Cases A,C and D which are mismatching cases,as well as the matching cases of A and B.Furthermore,the results indicate that the pressure signal measurements for such analysis should be within the bed away from the disengagement and sparger zones.

Fig.8.Radial profiles of the Kolmogorov entropy for Cases A and B of similar hydrodynamics with matching radial profiles of gas hold up at different axial levels,(a)H/D c=0.75,(b)H/D c=1.5,and(c)H/D c=1.75.

The percentage change in the average absolute relative difference(AARD)between Cases A and C was about 15.1%at H/Dc=0.75.In addition,the difference became relatively smaller when H/Dcchanged from 0.75 to 1.5.The percentage change in the average absolute relative difference(AARD)was about 13.5%at H/Dc=1.5.The same trend of non-similar or not close radial profiles of the Kolmogorov entropy was obtained at H/Dc=1.75 but with less deviation compared with Case A,in which the percentage change in the average absolute relative difference(AARD)was about 3.1%at H/Dc=0.75.The radial variations of the percentage change in the absolute relative difference(ARD)of the Kolmogorov entropy between Cases A and C at three axial heights above the distributor is shown in Fig.7(middle column),which shows that the radial variations of the ARD at different axial levels follow no uniform trends either radially or axially for all the local measurements of the Kolmogorov entropy.

The percentage change in the average absolute relative difference(AARD)between Cases A and D was about 13.9%at H/Dc=0.75.In addition,the difference became relatively smaller when H/Dcchanged from 0.75 to 1.5.The percentage change in the average absolute relative difference(AARD)was about 18.9%at H/Dc=1.5.The same trend(i.e.,radial profiles of the Kolmogorov entropy that were not close)was obtained at H/Dc=1.75 but with less deviation compared with Case A,in which the percentage change in the average absolute relative difference(AARD)was about 4.1%at H/Dc=0.75.The radial variations of the percentage change in the absolute relative difference(ARD)of the Kolmogorov entropy between Cases A and D at three axial levels above the distributor plate are shown in Fig.7(right column).The same nonuniform behavior of the radial variations of the ARD that was obtained in Fig.7(middle column)at different axial levels was also obtained in Fig.7(right column)at different radial and axial positions for all the local measurements of the Kolmogorov entropy.

7.Remarks

The chaotic scale-up approach for the gas–solid fluidized beds proposed by Schouten et al.[14]that is based on maintaining the same rate of information loss in terms of Kolmogorov entropy between the two scales has been assessed.We used the conditions of using our validated new mechanistic scale-up methodology that was based on matching the radial profiles of the gas holdup between the two fluidized beds to ensure the similarity in local hydrodynamics measured by advanced techniques.For these conditions,pressure gauge transducer measurements were performed at the wall and using a local pressure probe to measure the local pressure fluctuation at different radial and axial heights.The following conclusions have been found:

Fig.9.Radial profiles of the Kolmogorov entropy for Cases A and C of non-similar hydrodynamics with mismatching radial profiles of gas holdup at different axial levels,(a)H/D c=0.75,(b)H/D c=1.5,and(c)H/D c=1.75.

(1)When KE is close or matched in two scales or two different conditions with geometrical similarity of gas–solid fluidized beds,the details of local dimensionless hydrodynamic parameters will be similar as per the measurements reported by Zaid[6],Al-Dahhan et al.[17],Efhaima[7],and Efhaima and Al-Dahhan[8]using advanced measurement techniques of optical fiber probe utilization,radioactive particle tracking(RPT),gamma ray tomography(CT),and gamma ray densitometry(GRD).

(2)When KE is not matched or is not close to each other for the two scales and conditions with geometrical similarity of gas–solid fluidized beds,the detailed local dimensionless hydrodynamic parameters will not be similar.

(3)The measurement of the pressure signal for estimating the KE

for scale-up should be within the bed away from the freeboard and sparger regions to ensure the hydrodynamic similarity in scale-up by matching KE.

Nomenclature

Dcinside column diameter,m

dpparticle diameter,μm

H static bed height,m

L column height,m

P operating pressure,k Pa

R radius of the column,m

r radial position,m

T operating temperature,K

Ugsuperficial gas velocity,m·s?1

Umfminimum fluidization velocity,m·s?1

ε gas holdup

μ viscosity/gas viscosity,kg·m?1·s?1

ρ density,kg·m?3

ρffluid density or gas density,kg·m?3

ρssolid particle density,kg·m?3

φ sphericity

Subscripts

g gas

p particle

s solid

Acknowledgments

The authors would like to thank the Multiphase Reactors Engineering and Applications Laboratory(mReal)for funding and support.

Fig.10.Radial profiles of the Kolmogorov entropy for Cases A and D of non-similar hydrodynamics with mismatching radial profiles of gas holdup at different axial levels,(a)H/D c=0.75,(b)H/D c=1.5,and(c)H/D c=1.75.